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Answers to Selected Problems
Answers to Selected Problems
Chapter 1
Section 2 , page 12 (a) (b) (c)
( h e f ,k2et, k3ef) (kle‘, k2e-2t,k3) ( k l e f ,k2e-2f, h e Z f ) 6. A = diag {ai, ..., a,) and ai < 0, i = 1, . . ., n. 8. (b) Any solutions u, v such that u ( 0 ) and u(0) are independent vectors. 2.
Chapter 2
Page 27
1. F ( z )
dx2 dP
m-
=
- K x : V ( r ) = K l l x l l * , x ER2 2
=
-grad V
~
=
-Kx
“hlost” initial conditions means the set of collinear with x .
2.
(a) with V ( x ,y)
=
23
2y3 3
v) E R2 X R2 such that u is not
(2,
- - - - and (c) with V ( x ,y ) 3
7. Hint: Use (4) Section 6.
=
22
2
344
ANSWERS TO SELECTED PROBLEMS
Chapter 3 Section 3, page 54
1. (a) c ( t ) = 0, y(t) = 3eZ1
4. All eigenvalues are positive.
6.
(b) b
>0
Section 4 , page 60 1.
+
(d) c = 3et cos 2t 9eg sin 2t y = 3ef sin 21 - 9e' cos 2t.
Chapter 4 Section 1, page 65
2. dim E = dim EC and dim F 3. F > RCR
2
dim FR
Section 2, page 69
1. (a) Basis for E is given by (0, -fl,fi)and (1, -2, -1).
Section 3, page 73
-a, a),(1, -2,
Introduce the new basis (1,0 , O ) , (0, (yl, YZ, y3) related to the old by XI 22
xa
+
= YI Ys, = -fi y2 2ys, = fi y2 y3.
-
-
- 1), and new coordinates
345
ANSWERS TO SELECTED PROBLEMS
In the new coordinates the differential equation becomes Y: = Y l y; = -fi y31 y; = fi y2.
The general solution is yl = Cet, yz = A cos(fi 1) y3 = -B c o s ( f i t )
Thcreforc
+ B sin(fi t ) , + A s i n ( a t).
+
= Ce' - B cos(fi t ) A sin(V2 t), .rz= (2B- A a ) cos(* t ) - ( B f i 2 A ) s in (fi t ) , 23 = ( B A f i ) c o s ( f i t) ( B f i - A ) s i n ( f i t). $1
+
+
+
(The authors solved this problem in only two days.)
Chapter 5
Section 2 , page 81 3. A = l , B = & 4. (a) fi ( b) 6 . (a) and (d)
t
(c)
1
(d)
4
Section 3, page 87
3.
5
Hint: Note ITXI - ITYI --I T y I i f y = - - . 1x1
IYI
1x1
4.
(a) The norm is 1. 7. Hint: Use geometric series. W
1
1-0
1-
C x i ---
13.
5
for O < s < l ,
withx=(II-TI(.
Hint: Show that all the terms in the power series for
eA leave
E invariant.
346
ANSWER8 TO BELECTED PROBLEMS
Section 4 , page 97
1. (a) (b)
z(t) y(t) ~ ( t y(t)
= ( K 2 - tK1)e2', = Kle2'. = ) e2'(K1cos t - K 2 sin t ) , = e2'(K2 cos t Kt sin t ) .
+
+
2. (a) z ( t ) = (2t l)e2', y ( t ) = -2e21. (b) z ( t ) = 2e2' sin t , y ( t ) = -2e2' cos 1. 4. Hint: Consider A restricted to eigenspaces of X and use result of Problem 3. 9. (a) sink (b) source (c) source (d) none of these ( f ) none of these 10. (a) Only if a < - 2 are there any values of such k and in this case for
k>-. (b) No values of k .
14.
Hint: There is a real eigenvalue. Study T on its eigenspace.
Section 5 , page 102
1. (a) (b) (c)
+ sin t ] - + e4%. e4 z ( t ) = -f9[4t + 1) + - + e4Ik. 16 s ( 1 ) = A cos t + B sin t . y ( t ) = -A sin.t + B cos t + 21. x ( t ) = A[-4
cos t
Section 6 , page 107
+
( b ) s ( l ) = -ef+ e2r-2. 3. (a) c o s f i 2 , sin* 1 (b) e x p f i t , exp -fi t 4. Hint: Check cases ( a ) , ( b ) , (c) of the t,heorem. 8. a = 0, b > 0; period is 4 / 2 r . 2.
(a)
s ( l ) = cos 21.
Chapter 6 Section 2, page 120
1. (a)
'1
Gcneralizcd l-eigenspace spanned by (1, 0 ) , (0, 1 ) ;
s = [ o1
J, 0
N=[O
0 0
347
INSWERS TO SELECTED PROBLEMS
(b) Generalized 1-eigenspace spanned by (1,O) ; generalized ( - 1)-cigcnspace spanned by (1, 2) ;
c' 'I,
s=
0
1v =
-1
[;
;I.
2. If the rth power of the mat,rix is [b,,], then b , j = 0 for i 3. The only eigenvalue is 0.
+ r ( r = 1, 2, . . . ) .
+
iV decomposition. 5 . Consider the S 6. A prcservrs each generalized eigenspace EX; hence it suffices to consider the restrictions of A and T to Ex. If T = S N , then S Ex = X I which commutes with A . Thus S and T both commute with A ; so therefore does N = T - S. S. Use thr Cayley-Hamilton theorem. 1.5. Consider bases of the kernel and the image.
+
Section 3, page 126 1.
Canonical forms:
3. Assumc that N is in nilpotent canonical form. Let b denote the number of blocks and s the maximal number of rows in a block. Then bs 5 n ; also b = n - T and s
4.
5 k.
Similar pairs are ( a ) , (d) and (b) , (c)
Section 4 , page 132
1.
(a)
4. F o r n
[Z 0
=
-i O ]
(c)
[ I0f i l + i
3:
o o c
3
348
ANSWERS TO SELECTED PROBLEMS
6. If Ax = p t , 5 # 0, then 0 = q ( A ) z = q(p)x. 8. Show that A and A t have the same Jordan form if A is a complex matrix, and the same real canonical form if A is real. Section 5, page 136
+
1. (a) Let every eigenvalue have real part < - b with b > a > 0. Let A = S N with S semisimple and N nilpotent. In suitable coordinates 1) ers )I 5 e-lb, )I etx 11 5 C P . Then 11 elA 11 Ic e f b l n ,and so eta I( el* I( + O as t 4 00. Lets>Obesolargethateto((etA (1 < 1 f o r t k s . P u t k = min(((etA1l-I) for 0 5 t I s. 2. If z is an eigenvector belonging to an eigenvalue with nonzero real part, then the solution etAxis not periodic. If ib, iC are pure imaginary eigenvalues, b # f c , and z , tu E Cn are corresponding eigenvectors, then the real part of etA( z w) is a nonperiodic solution.
+
Section 6, page 141
e-l. (a) In ( 7 ) , A = B s y o ) = -D.
1. s ( l )
2.
=
=
0. Hence s(0) = C, s'(0)
=
D,
s(*)(O) = -c,
Chapter 7 Section I , page 150
3. (a) Use etBelA = Section 2, page 153 2. Use the theorem of this section and Theorems 1 and 2 of Section 1. 3. Use Problem 2.
Section 3, page J57
1. (a) dense, open ( e ) open
(b) dense (f) open
(c) dense, open (9) dense, open
349
ANSWERS TO SELECTED PROBLEMS
Chapter 8
Page 177
1.
(a)
f(s) =
5
+ 2.
uo(t)
=
2,
+ / t f ( u ~ ( s ) )ds 2 + 4t,
ul(t) = 2 =
~ ( t =) 2
0
=
2
+ /'4ds 0
+ /' (4 + 4s) ds = 2 + 4t + 2t2,
l' 0
ua(t) = 2 +
(4
+ 4s + 2s2) ds = 2 + 4t + 2t2 + 2t3 -3.
By induction + - t") - 2 .
.*.
n!
Hence
z ( t ) = limu,(t) = 4et - 2. n-m
u,(t) = 0
for all n: Hence z ( t ) = 0. 4.
(c)
z(t) =
(a)
1
(b) 5.
'
t 3 .
f ( z ) - '(')
(2-01
(c) 1 (a) For 0
'
+ m as 2 + 0; no Lipschitz constant.
< c < Blet x(t) =
1"'
&(t
OltSc
- c)3,
c
5 1 58.
350
ANSWERS TO SELECTED PROBLEMS
Chapter 9 Section 1, page 185
2. For example, f(r) = -x3 (z E R). 3. H i n t : Use a special inner product on Rn. Compute the rate of change of I x ( t ) I* where x ( t ) is a solution such that s(0) is (the real part of) an eigenvector for Df(0)having positive real part; take z(0) very small. 4. Use ( b ) of the theorem of Section 1. Section 2 , page 191
2. (a), (b), (e) 3. H i n t : Look at the Jordan form of A . It suffices to consider an elementary .Jordan block. Section 3, page 199 1. x2
+ y2 is a strict Liapunov function.
S. V-l[O, c] is positively invariant. The w-limit set of any point of V-l[O, c] consists entirely of equilibria in V-'[O, c]; hence it is just 2 . Section 4, page 204
2. Let x' = -grad V ( z ) .Then V decreases along trajectories, so that V is constant on a recurrent trajectory. Hence, a recurrent trajectory consists entirely of equilibrium points, and so is a constant. 3. (a) Each set V-l( - m , c] is positively invariant. (h) Use Theorem 3. Section 5, page 209
1. H i n t : Find eigrnvcctors. 2. Let Ax = Xx, A y = py, X # Xp-'(z, y), and Xp-' # 1. 5 . A x = grad $ ( x , Ax).
p, p #
0. Then
(2,
y ) = p-l(x, A y ) = p - l ( A z , y) =
351
ANSWERS TO SELECTED PROBLEMS
Chapter 10
Section I , page 215
x =
k,
y = vc.
Section 3, page 226
1. Every solution is periodic! Hint : If ( x( t ), y ( t ) ) is a solution, so is ( - x ( - t ) ,
Y( - t ) 1.
Section 4 , page 228 1. p = - 2 ,
p=-lf21/7.
Section 5, page 237
x =
iL,
y = vc
Chapter 11
Section 1, page 241
1. H i n t : If the limit set L is not connected, find disjoint open sets Ul, lT2 containing L. Then find a bounded sequence of points x , on the trajectory with X,
4.
t ui, 2,
[Jz.
H i n t : Every solution is periodic.
352
ANSWERS M SELECTED PROBLEMS
Section 3, page 247
2. Hint: Apply Proposition 2. 4. Hints: (a) If x is not an equilibrium, take a local section at Problem 2 of Section 1.
1.
(b) See
Section 4 , page 249 2. H i n t : Let y E y. Take a local section at y and apply Proposition 1 of the previous section.
Section 5, page 253
+
2. Hints: (a) Use Poincar6-Bendixson. (b) Do the problem for 2 n 1 closed orbits; use induction on n. 5. Hint: Let U be the region bounded by a closed orbit y o f f . Then g is transverse to the boundary y of U . Apply Poincar6-Bendixson. Chapter 13 Section 1 , page 278 (a) Hint: Show that the given condition is equivalent to the existence of an with I a I < 1. Apply Theorem 2. eigenvalue a of & ( x ) Section 3, page 285
3. Hint: If v is periodic of period A, then so is rv for all r > 0. 5. Hints: (a) Do the problem first in case p is zero and g is linear. Then use Taylor’s formula for the general case. (b) Apply the result in (a) after taking a local section. Chapter 15 Section 2, page 303
2. This is pretty trivial. Since 2’ is the Ct function f, then x is Cr+l.
353
ANSWERS TO SELECTED PROBLEMS
Chapter 16 Section 1 , page 309 1. Hint: If B is close to A , each eigenvalue of B having negative real part will
be close t o a similar eigenvalue A of A . Arguing as in the proof that 81is open in Theorem 1 of Chapter 7, Section 3, show that the sum of the multiplicities of these eigenvalues p , of B near A equals the multiplicity of A. Then show that bases for the generalized eigenspaces of the p , can be chosen near corresponding bases for A.
Section 3, page 318 1.
A.
+
Suppose Df(0)has 0 as a n eigenvalue, let g.(z) = f(z) cz, t # 0.For 1 t 1 sufficiently small, one of gWe,g. will be a saddle and the other a source or sink; hence j cannot have the same phase portrait as both g-e and g.. If Df(0)has +hi, A > 0, as a n eigenvalue, then g-. is a sink and g+. is a source. Hint: First consider the case where e l A is a contraction or expansion. Then use Problem 1 of Section 1.
Chapter 1
Section 2 , page 12 (a) (b) (c)
( h e f ,k2et, k3ef) (kle‘, k2e-2t,k3) ( k l e f ,k2e-2f, h e Z f ) 6. A = diag {ai, ..., a,) and ai < 0, i = 1, . . ., n. 8. (b) Any solutions u, v such that u ( 0 ) and u(0) are independent vectors. 2.
Chapter 2
Page 27
1. F ( z )
dx2 dP
m-
=
- K x : V ( r ) = K l l x l l * , x ER2 2
=
-grad V
~
=
-Kx
“hlost” initial conditions means the set of collinear with x .
2.
(a) with V ( x ,y)
=
23
2y3 3
v) E R2 X R2 such that u is not
(2,
- - - - and (c) with V ( x ,y ) 3
7. Hint: Use (4) Section 6.
=
22
2
344
ANSWERS TO SELECTED PROBLEMS
Chapter 3 Section 3, page 54
1. (a) c ( t ) = 0, y(t) = 3eZ1
4. All eigenvalues are positive.
6.
(b) b
>0
Section 4 , page 60 1.
+
(d) c = 3et cos 2t 9eg sin 2t y = 3ef sin 21 - 9e' cos 2t.
Chapter 4 Section 1, page 65
2. dim E = dim EC and dim F 3. F > RCR
2
dim FR
Section 2, page 69
1. (a) Basis for E is given by (0, -fl,fi)and (1, -2, -1).
Section 3, page 73
-a, a),(1, -2,
Introduce the new basis (1,0 , O ) , (0, (yl, YZ, y3) related to the old by XI 22
xa
+
= YI Ys, = -fi y2 2ys, = fi y2 y3.
-
-
- 1), and new coordinates
345
ANSWERS TO SELECTED PROBLEMS
In the new coordinates the differential equation becomes Y: = Y l y; = -fi y31 y; = fi y2.
The general solution is yl = Cet, yz = A cos(fi 1) y3 = -B c o s ( f i t )
Thcreforc
+ B sin(fi t ) , + A s i n ( a t).
+
= Ce' - B cos(fi t ) A sin(V2 t), .rz= (2B- A a ) cos(* t ) - ( B f i 2 A ) s in (fi t ) , 23 = ( B A f i ) c o s ( f i t) ( B f i - A ) s i n ( f i t). $1
+
+
+
(The authors solved this problem in only two days.)
Chapter 5
Section 2 , page 81 3. A = l , B = & 4. (a) fi ( b) 6 . (a) and (d)
t
(c)
1
(d)
4
Section 3, page 87
3.
5
Hint: Note ITXI - ITYI --I T y I i f y = - - . 1x1
IYI
1x1
4.
(a) The norm is 1. 7. Hint: Use geometric series. W
1
1-0
1-
C x i ---
13.
5
for O < s < l ,
withx=(II-TI(.
Hint: Show that all the terms in the power series for
eA leave
E invariant.
346
ANSWER8 TO BELECTED PROBLEMS
Section 4 , page 97
1. (a) (b)
z(t) y(t) ~ ( t y(t)
= ( K 2 - tK1)e2', = Kle2'. = ) e2'(K1cos t - K 2 sin t ) , = e2'(K2 cos t Kt sin t ) .
+
+
2. (a) z ( t ) = (2t l)e2', y ( t ) = -2e21. (b) z ( t ) = 2e2' sin t , y ( t ) = -2e2' cos 1. 4. Hint: Consider A restricted to eigenspaces of X and use result of Problem 3. 9. (a) sink (b) source (c) source (d) none of these ( f ) none of these 10. (a) Only if a < - 2 are there any values of such k and in this case for
k>-. (b) No values of k .
14.
Hint: There is a real eigenvalue. Study T on its eigenspace.
Section 5 , page 102
1. (a) (b) (c)
+ sin t ] - + e4%. e4 z ( t ) = -f9[4t + 1) + - + e4Ik. 16 s ( 1 ) = A cos t + B sin t . y ( t ) = -A sin.t + B cos t + 21. x ( t ) = A[-4
cos t
Section 6 , page 107
+
( b ) s ( l ) = -ef+ e2r-2. 3. (a) c o s f i 2 , sin* 1 (b) e x p f i t , exp -fi t 4. Hint: Check cases ( a ) , ( b ) , (c) of the t,heorem. 8. a = 0, b > 0; period is 4 / 2 r . 2.
(a)
s ( l ) = cos 21.
Chapter 6 Section 2, page 120
1. (a)
'1
Gcneralizcd l-eigenspace spanned by (1, 0 ) , (0, 1 ) ;
s = [ o1
J, 0
N=[O
0 0
347
INSWERS TO SELECTED PROBLEMS
(b) Generalized 1-eigenspace spanned by (1,O) ; generalized ( - 1)-cigcnspace spanned by (1, 2) ;
c' 'I,
s=
0
1v =
-1
[;
;I.
2. If the rth power of the mat,rix is [b,,], then b , j = 0 for i 3. The only eigenvalue is 0.
+ r ( r = 1, 2, . . . ) .
+
iV decomposition. 5 . Consider the S 6. A prcservrs each generalized eigenspace EX; hence it suffices to consider the restrictions of A and T to Ex. If T = S N , then S Ex = X I which commutes with A . Thus S and T both commute with A ; so therefore does N = T - S. S. Use thr Cayley-Hamilton theorem. 1.5. Consider bases of the kernel and the image.
+
Section 3, page 126 1.
Canonical forms:
3. Assumc that N is in nilpotent canonical form. Let b denote the number of blocks and s the maximal number of rows in a block. Then bs 5 n ; also b = n - T and s
4.
5 k.
Similar pairs are ( a ) , (d) and (b) , (c)
Section 4 , page 132
1.
(a)
4. F o r n
[Z 0
=
-i O ]
(c)
[ I0f i l + i
3:
o o c
3
348
ANSWERS TO SELECTED PROBLEMS
6. If Ax = p t , 5 # 0, then 0 = q ( A ) z = q(p)x. 8. Show that A and A t have the same Jordan form if A is a complex matrix, and the same real canonical form if A is real. Section 5, page 136
+
1. (a) Let every eigenvalue have real part < - b with b > a > 0. Let A = S N with S semisimple and N nilpotent. In suitable coordinates 1) ers )I 5 e-lb, )I etx 11 5 C P . Then 11 elA 11 Ic e f b l n ,and so eta I( el* I( + O as t 4 00. Lets>Obesolargethateto((etA (1 < 1 f o r t k s . P u t k = min(((etA1l-I) for 0 5 t I s. 2. If z is an eigenvector belonging to an eigenvalue with nonzero real part, then the solution etAxis not periodic. If ib, iC are pure imaginary eigenvalues, b # f c , and z , tu E Cn are corresponding eigenvectors, then the real part of etA( z w) is a nonperiodic solution.
+
Section 6, page 141
e-l. (a) In ( 7 ) , A = B s y o ) = -D.
1. s ( l )
2.
=
=
0. Hence s(0) = C, s'(0)
=
D,
s(*)(O) = -c,
Chapter 7 Section I , page 150
3. (a) Use etBelA = Section 2, page 153 2. Use the theorem of this section and Theorems 1 and 2 of Section 1. 3. Use Problem 2.
Section 3, page J57
1. (a) dense, open ( e ) open
(b) dense (f) open
(c) dense, open (9) dense, open
349
ANSWERS TO SELECTED PROBLEMS
Chapter 8
Page 177
1.
(a)
f(s) =
5
+ 2.
uo(t)
=
2,
+ / t f ( u ~ ( s ) )ds 2 + 4t,
ul(t) = 2 =
~ ( t =) 2
0
=
2
+ /'4ds 0
+ /' (4 + 4s) ds = 2 + 4t + 2t2,
l' 0
ua(t) = 2 +
(4
+ 4s + 2s2) ds = 2 + 4t + 2t2 + 2t3 -3.
By induction + - t") - 2 .
.*.
n!
Hence
z ( t ) = limu,(t) = 4et - 2. n-m
u,(t) = 0
for all n: Hence z ( t ) = 0. 4.
(c)
z(t) =
(a)
1
(b) 5.
'
t 3 .
f ( z ) - '(')
(2-01
(c) 1 (a) For 0
'
+ m as 2 + 0; no Lipschitz constant.
< c < Blet x(t) =
1"'
&(t
OltSc
- c)3,
c
5 1 58.
350
ANSWERS TO SELECTED PROBLEMS
Chapter 9 Section 1, page 185
2. For example, f(r) = -x3 (z E R). 3. H i n t : Use a special inner product on Rn. Compute the rate of change of I x ( t ) I* where x ( t ) is a solution such that s(0) is (the real part of) an eigenvector for Df(0)having positive real part; take z(0) very small. 4. Use ( b ) of the theorem of Section 1. Section 2 , page 191
2. (a), (b), (e) 3. H i n t : Look at the Jordan form of A . It suffices to consider an elementary .Jordan block. Section 3, page 199 1. x2
+ y2 is a strict Liapunov function.
S. V-l[O, c] is positively invariant. The w-limit set of any point of V-l[O, c] consists entirely of equilibria in V-'[O, c]; hence it is just 2 . Section 4, page 204
2. Let x' = -grad V ( z ) .Then V decreases along trajectories, so that V is constant on a recurrent trajectory. Hence, a recurrent trajectory consists entirely of equilibrium points, and so is a constant. 3. (a) Each set V-l( - m , c] is positively invariant. (h) Use Theorem 3. Section 5, page 209
1. H i n t : Find eigrnvcctors. 2. Let Ax = Xx, A y = py, X # Xp-'(z, y), and Xp-' # 1. 5 . A x = grad $ ( x , Ax).
p, p #
0. Then
(2,
y ) = p-l(x, A y ) = p - l ( A z , y) =
351
ANSWERS TO SELECTED PROBLEMS
Chapter 10
Section I , page 215
x =
k,
y = vc.
Section 3, page 226
1. Every solution is periodic! Hint : If ( x( t ), y ( t ) ) is a solution, so is ( - x ( - t ) ,
Y( - t ) 1.
Section 4 , page 228 1. p = - 2 ,
p=-lf21/7.
Section 5, page 237
x =
iL,
y = vc
Chapter 11
Section 1, page 241
1. H i n t : If the limit set L is not connected, find disjoint open sets Ul, lT2 containing L. Then find a bounded sequence of points x , on the trajectory with X,
4.
t ui, 2,
[Jz.
H i n t : Every solution is periodic.
352
ANSWERS M SELECTED PROBLEMS
Section 3, page 247
2. Hint: Apply Proposition 2. 4. Hints: (a) If x is not an equilibrium, take a local section at Problem 2 of Section 1.
1.
(b) See
Section 4 , page 249 2. H i n t : Let y E y. Take a local section at y and apply Proposition 1 of the previous section.
Section 5, page 253
+
2. Hints: (a) Use Poincar6-Bendixson. (b) Do the problem for 2 n 1 closed orbits; use induction on n. 5. Hint: Let U be the region bounded by a closed orbit y o f f . Then g is transverse to the boundary y of U . Apply Poincar6-Bendixson. Chapter 13 Section 1 , page 278 (a) Hint: Show that the given condition is equivalent to the existence of an with I a I < 1. Apply Theorem 2. eigenvalue a of & ( x ) Section 3, page 285
3. Hint: If v is periodic of period A, then so is rv for all r > 0. 5. Hints: (a) Do the problem first in case p is zero and g is linear. Then use Taylor’s formula for the general case. (b) Apply the result in (a) after taking a local section. Chapter 15 Section 2, page 303
2. This is pretty trivial. Since 2’ is the Ct function f, then x is Cr+l.
353
ANSWERS TO SELECTED PROBLEMS
Chapter 16 Section 1 , page 309 1. Hint: If B is close to A , each eigenvalue of B having negative real part will
be close t o a similar eigenvalue A of A . Arguing as in the proof that 81is open in Theorem 1 of Chapter 7, Section 3, show that the sum of the multiplicities of these eigenvalues p , of B near A equals the multiplicity of A. Then show that bases for the generalized eigenspaces of the p , can be chosen near corresponding bases for A.
Section 3, page 318 1.
A.
+
Suppose Df(0)has 0 as a n eigenvalue, let g.(z) = f(z) cz, t # 0.For 1 t 1 sufficiently small, one of gWe,g. will be a saddle and the other a source or sink; hence j cannot have the same phase portrait as both g-e and g.. If Df(0)has +hi, A > 0, as a n eigenvalue, then g-. is a sink and g+. is a source. Hint: First consider the case where e l A is a contraction or expansion. Then use Problem 1 of Section 1.